PSI - Issue 37

Arvid Trapp et al. / Procedia Structural Integrity 37 (2022) 622–631 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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2.4. Non-Gaussian models (kurtosis control) Most real-world random vibration phenomena, e.g. recorded in-service loading on vehicles, usually differ significantly from the stationary Gaussian assumption (Eq. (1). This is due to changing operational, environmental or excitational conditions which result in diverse vibration with varying intensities. The kurtosis has not only proven to be a popular descriptor for the non-Gaussianity of a load series, but it has also been the central parameter to approximate realistic loading by kurtosis control. Kurtosis control generates load series based on a desired PSD and kurtosis value. However different mechanisms can be applied for this goal and still, it has not been established a clear path on which control algorithm is best suitable for any given circumstances. In the following two popular algorithms are introduced which can be categorized into stationary non-Gaussian (HP) and non-stationary non-Gaussian (CN) random loading.

(a)

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Figure 2: (b) Hermite polynomials (HP) and (c) carrier-noise (CN) realization with data length of (a) same PSD and kurtosis

Nonlinear transformation via Hermite polynomials (HP) – stationary non-Gaussian This approach manipulates an initial Gaussian realization ( ) , defined by the desired PSD 2 ( ) ( ), using a nonlinear transformation scheme (Trapp and Wolfsteiner, 2019; Winterstein, 1988). A normalization of the Gaussian realization to = 0 and 2 = is necessary in order to reliably achieve a given output standard deviation and kurtosis value , generated by the nonlinear transformation {⋅} . The correction factor provides this necessary normalization which is then scaled for the output load series ( ) . ℎ̅ 4 = √1+1 5( −3)−1 18 ; = √1+ 1 6ℎ̅ 42 ; ( ) = { ( )} = ( ( ) + ℎ̅ 4 (( ( ) ) 3 − 3 ( ( ) ))) (8) Carrier-Noise (CN) – non-stationary non-Gaussian The intensity of vibration loading may regularly vary due to external influences. This varying variance 2 ( ) can be approximated by a low-frequent signal ( ) that modulates the original load series ( ) with given PSD in time. To realize a modulated load series ( ) = ( ) ( ) with a desired kurtosis , there are different approaches (Cui and Liu Bin, 2021; Kihm et al., 2015; Trapp et al., 2019) available for the modulating signal ( ) . Hermite polynomial (HP) and carrier-noise (CN) realization Both kurtosis control mechanisms are used in the following to generate non-Gaussian realizations of same PSD and kurtosis. They function throughout this paper as exemplary non-Gaussian data. The PSD is synthetically defined by a constant-level colored-noise PSD (Figure 2a). Applying the introduced kurtosis control algorithms, the HP (Figure 2b) and the CN (Figure 2c) realizations reproduce a kurtosis value of = 0 . 3. On the transmission of non-Gaussian random loading through linear structures 3.1. Visualizing trispectra As commented in the previous Sections 2.3 and 2.4, the multi-dimensionality of the trispectrum makes it challenging to estimate, interpret, and visualize. For a meaningful depiction of this fourth- order moment’s frequency domain decomposition several visualization approaches have been proposed. In the following these approaches are briefly compared. On this basis, an interactive and adaptive visualization approach is proposed. The complex-valued